| L(s) = 1 | + (−0.950 + 1.04i)2-s + (1.70 − 0.282i)3-s + (−0.194 − 1.99i)4-s + (0.762 − 2.24i)5-s + (−1.32 + 2.05i)6-s + (0.172 − 0.224i)7-s + (2.27 + 1.68i)8-s + (2.84 − 0.963i)9-s + (1.62 + 2.93i)10-s + (0.890 + 0.0583i)11-s + (−0.894 − 3.34i)12-s + (−0.972 + 1.10i)13-s + (0.0715 + 0.393i)14-s + (0.669 − 4.05i)15-s + (−3.92 + 0.775i)16-s + (−3.08 − 3.08i)17-s + ⋯ |
| L(s) = 1 | + (−0.671 + 0.740i)2-s + (0.986 − 0.162i)3-s + (−0.0974 − 0.995i)4-s + (0.340 − 1.00i)5-s + (−0.542 + 0.840i)6-s + (0.0651 − 0.0848i)7-s + (0.802 + 0.596i)8-s + (0.946 − 0.321i)9-s + (0.514 + 0.926i)10-s + (0.268 + 0.0176i)11-s + (−0.258 − 0.966i)12-s + (−0.269 + 0.307i)13-s + (0.0191 + 0.105i)14-s + (0.172 − 1.04i)15-s + (−0.981 + 0.193i)16-s + (−0.748 − 0.748i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.343i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 + 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.52901 - 0.271177i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.52901 - 0.271177i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.950 - 1.04i)T \) |
| 3 | \( 1 + (-1.70 + 0.282i)T \) |
| good | 5 | \( 1 + (-0.762 + 2.24i)T + (-3.96 - 3.04i)T^{2} \) |
| 7 | \( 1 + (-0.172 + 0.224i)T + (-1.81 - 6.76i)T^{2} \) |
| 11 | \( 1 + (-0.890 - 0.0583i)T + (10.9 + 1.43i)T^{2} \) |
| 13 | \( 1 + (0.972 - 1.10i)T + (-1.69 - 12.8i)T^{2} \) |
| 17 | \( 1 + (3.08 + 3.08i)T + 17iT^{2} \) |
| 19 | \( 1 + (0.658 - 3.31i)T + (-17.5 - 7.27i)T^{2} \) |
| 23 | \( 1 + (-6.76 + 5.19i)T + (5.95 - 22.2i)T^{2} \) |
| 29 | \( 1 + (-2.30 + 4.66i)T + (-17.6 - 23.0i)T^{2} \) |
| 31 | \( 1 + (2.39 + 1.38i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.25 + 6.29i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (4.82 - 3.69i)T + (10.6 - 39.6i)T^{2} \) |
| 43 | \( 1 + (0.0912 - 1.39i)T + (-42.6 - 5.61i)T^{2} \) |
| 47 | \( 1 + (1.23 - 4.62i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (0.782 - 1.17i)T + (-20.2 - 48.9i)T^{2} \) |
| 59 | \( 1 + (-3.63 + 10.7i)T + (-46.8 - 35.9i)T^{2} \) |
| 61 | \( 1 + (-3.14 - 1.55i)T + (37.1 + 48.3i)T^{2} \) |
| 67 | \( 1 + (-0.852 - 13.0i)T + (-66.4 + 8.74i)T^{2} \) |
| 71 | \( 1 + (0.728 + 1.75i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (4.44 - 10.7i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (-0.828 - 0.222i)T + (68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (-3.03 + 1.02i)T + (65.8 - 50.5i)T^{2} \) |
| 89 | \( 1 + (7.90 - 3.27i)T + (62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 + (-4.93 + 2.84i)T + (48.5 - 84.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.25158481100434482267054923845, −9.374784500369101020389195580249, −8.925469064143409997231116396883, −8.198267123463489056287191881020, −7.23568064954908293920069969734, −6.42196374579656037401706861284, −5.09165736690742607436380673247, −4.27278229301215392680802054472, −2.39040959714835400594080915552, −1.10091334663242453883945144677,
1.74407352005750554225443560371, 2.83979496198500493984371639134, 3.55082125453278555619771197422, 4.88370006192047371013276823896, 6.78175740404578520899185608408, 7.24171785231406268310391683704, 8.493769400895670580570114877224, 8.997010958886225825411110291488, 9.977063954010187054951215508866, 10.60137867670116269037074304929
